Solve for X: Rectangle with Perimeter 20 and Sides 4X+3 and 2X+1

Q: Why do I multiply each side expression by 2?

In a rectangle, opposite sides are equal. So you have two sides of length 4x+3 and two sides of length 2x+1. The perimeter includes all four sides!

Q: Can I just add the two different expressions to get the perimeter?

No! That would only give you half the perimeter. You need all four sides: (4x+3) + (2x+1) + (4x+3) + (2x+1) = 20.

Q: How do I know which expression goes with which side?

Look at the diagram carefully! The horizontal sides are labeled 4x+3, and the vertical sides are labeled 2x+1. Opposite sides in rectangles are always equal.

Rectangle Perimeter with Variable Expressions

Look at the following rectangle:

The perimeter of the rectangle is 20.

Calculate the value of X.

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:04 First, let's find the value of X.

00:09 Remember, in a rectangle, opposite sides are the same length.

00:35 The perimeter is the total distance around the rectangle.

00:43 Now, plug in the numbers you have and solve for X.

01:15 Combine all terms that are alike.

01:30 Next, we want to get X all by itself.

01:43 And there you have it, that's the solution!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Look at the following rectangle:

The perimeter of the rectangle is 20.

Calculate the value of X.

Step-by-step solution

Since in a rectangle every pair of opposite sides are equal, we can claim that:

$AD=BC=2x+1$

$AB=CD=4x+3$

Since the perimeter of the rectangle is equal to 20, we can substitute the data into the formula:

$20=2x+1+4x+3+2x+1+4x+3$

$20=12x+8$

$20-8=12x$

$12=12x$

Let's divide both sides by 12:

$1=x$

Final Answer

$1$

Key Points to Remember

Essential concepts to master this topic

Rectangle Rule: Opposite sides are equal and perimeter equals sum of all four sides
Technique: Substitute expressions: $2(4x+3) + 2(2x+1) = 20$
Check: When x=1: sides are 7 and 3, perimeter is 2(7)+2(3)=20 ✓

Common Mistakes

Avoid these frequent errors

Adding each side expression only once
Don't use 4x+3+2x+1=20 as your equation = wrong perimeter formula! This only adds two sides, not all four. Always remember that perimeter includes each side twice: 2(4x+3)+2(2x+1)=20.

Practice Quiz

Test your knowledge with interactive questions

Look at the rectangle ABCD below.

Side AB is 6 cm long and side BC is 4 cm long.

What is the area of the rectangle?

FAQ

Everything you need to know about this question

Why do I multiply each side expression by 2?

In a rectangle, opposite sides are equal. So you have two sides of length 4x+3 and two sides of length 2x+1. The perimeter includes all four sides!

Can I just add the two different expressions to get the perimeter?

No! That would only give you half the perimeter. You need all four sides: (4x+3) + (2x+1) + (4x+3) + (2x+1) = 20.

What if I get a negative value for x?

Check your work! In geometry problems, side lengths must be positive. If x gives negative side lengths, you likely made an error in setting up or solving the equation.

How do I know which expression goes with which side?

Look at the diagram carefully! The horizontal sides are labeled 4x+3, and the vertical sides are labeled 2x+1. Opposite sides in rectangles are always equal.

Should I simplify 12x + 8 = 20 differently?

The standard method is perfect: subtract 8 from both sides to get 12x = 12, then divide by 12. This gives x = 1, which you can easily verify!

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